# Internal moment in the hull of a pressure vessel

This question is related to the course structural analysis. As part of our exam grade every student has been given different multiple homework assignments which we have to solve.

One of the problems I have been given is the following;

Given a cylinder with radius $R$ and internal pressure $p$, find the maximum bending moment in the cross section. Provide a free body diagram to support your answer.

The bending moment which is meant is the internal moment created in the hull of the cylinder when you take a cut out of cross section which is a circle obviously.

At this point I have only been able to proof that the internal shear force $V = 0$ and that the internal normal force $N$ must be equal to $N = p \cdot R$. Which can be proven as follows using the free body diagrams below.

$$2 \cdot N = R\cdot \int_0^\pi p \cdot sin(\theta) d\theta$$ $$Pd\theta = 2 \cdot N \cdot sin(0.5 \cdot d\theta) + 2\cdot V \cdot cos(0.5 \cdot d\theta)$$

Another way I have been looking at this problem is by making use of strains $\epsilon$ and stresses $\sigma$. If you assume the strain $\epsilon$ on the outside and inside of the cylinder are equal the stress $\sigma$ distribution over the thickness must be constant thus the moment $M$ must be 0. However the assumption of the strains being equal on the in and outside is something I am not sure about.

So what I'm looking for is a solution for the internal moment $M$ in the hull of the cylinder and if it is zero, a proof for this.

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I was thinking this might be modeled as a thin walled pressure vessel, but that would require you to know the thickness or the inner/outer radius independently. Is that all the information the question gives you? – Asad Saeeduddin Mar 25 '13 at 5:07